Difference between revisions of "Sequence"
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This naturally then generalises to [[Indexing set|indexing sets]] | This naturally then generalises to [[Indexing set|indexing sets]] | ||
+ | ==Notation== | ||
+ | To specify that the points of a sequence, the {{M|x_i}} are from a space, {{M|X}} we may write: | ||
+ | * {{M|1=(x_n)^\infty_{n=1}\subseteq X}} | ||
+ | This is an abuse of notation, as {{M|1=(x_n)^\infty_{n=1} }} is not a subset of {{M|X}}. It plays on: | ||
+ | * {{M|1=[(x_n)^\infty_{n=1}\subseteq X]\iff[x\in(x_n)_{n=1}^\infty\implies x\in X]}} | ||
+ | Note that the elements of {{M|1=(x_n)_{n=1}^\infty}} are ether: | ||
+ | * Elements of a [[Relation|relation]] (if we consider the sequence as a [[Function|mapping]]) or | ||
+ | ** So using this, {{M|1=x\in(x_n)_{n=1}^\infty}} may look like {{M|1=x=(a,b)}} (indicating {{M|1=f(a)=b}}) which is an [[Ordered pair]], not in {{M|X}} | ||
+ | * Elements of a [[Tuple|tuple]] (which is a generalisation of [[Ordered pair|ordered pairs]] where (usually) {{M|1=(a,b)=\{\{a\},\{a,b\}\} }} | ||
+ | ** So using this, {{M|1=x\in(x_n)_{n=1}^\infty}} may indeed look like {{M|1=x=\{\{a\},\{a,b\}\}\notin X}} | ||
+ | |||
+ | '''As such the notation {{M|1=(x_n)^\infty_{n=1}\subseteq X}} having no ''other'' sensible meaning''' is a notation to say that {{M|1=\forall i[x_i\in X]}} | ||
==Subsequence== | ==Subsequence== | ||
Given a sequence {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''<ref name="Analysis"/> as a sequence: | Given a sequence {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''<ref name="Analysis"/> as a sequence: | ||
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* [[Monotonic sequence]] | * [[Monotonic sequence]] | ||
* [[Bolzano-Weierstrass theorem]] | * [[Bolzano-Weierstrass theorem]] | ||
− | * [[Cauchy criterion for convergence]] | + | * [[Cauchy sequence]] (Alternatively: [[Cauchy criterion for convergence]]) |
* [[Convergence of a sequence]] | * [[Convergence of a sequence]] | ||
Revision as of 13:56, 9 July 2015
A sequence is one of the earliest and easiest definitions encountered, but I will restate it.
I was taught to denote the sequence {a1,a2,...} by {an}∞n=1 however I don't like this, as it looks like a set. I have seen the notation (an)∞n=1 and I must say I prefer it. This notation is inline with that of a tuple which is a generalisation of an ordered pair.
Contents
[hide]Definition
Formally a sequence (Ai)∞i=1 is a function[1][2], f:N→S where S is some set. For a finite sequence it is simply f:{1,...,n}→S. Now we can write:
- f(i):=Ai
This naturally then generalises to indexing sets
Notation
To specify that the points of a sequence, the xi are from a space, X we may write:
- (xn)∞n=1⊆X
This is an abuse of notation, as (xn)∞n=1 is not a subset of X. It plays on:
- [(xn)∞n=1⊆X]⟺[x∈(xn)∞n=1⟹x∈X]
Note that the elements of (xn)∞n=1 are ether:
- Elements of a relation (if we consider the sequence as a mapping) or
- So using this, x∈(xn)∞n=1 may look like x=(a,b) (indicating f(a)=b) which is an Ordered pair, not in X
- Elements of a tuple (which is a generalisation of ordered pairs where (usually) (a,b)={{a},{a,b}}
- So using this, x∈(xn)∞n=1 may indeed look like x={{a},{a,b}}∉X
As such the notation (xn)∞n=1⊆X having no other sensible meaning is a notation to say that ∀i[xi∈X]
Subsequence
Given a sequence (xn)∞n=1 we define a subsequence of (xn)∞n=1[2] as a sequence:
- k:N→N which operates on an n∈N with n↦kn:=k(n) where:
- kn is increasing, that means kn≤kn+1
We denote this:
- (xkn)∞n=1
See also
- Monotonic sequence
- Bolzano-Weierstrass theorem
- Cauchy sequence (Alternatively: Cauchy criterion for convergence)
- Convergence of a sequence
References
- Jump up ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
- ↑ Jump up to: 2.0 2.1 p11 - Analysis - Part 1: Elements - Krzysztof Maurin